TMD splitting functions and the corresponding evolution equation

We generalize the Catani Hautmann formalism to calculate the Pgq and Pqq transverse momentum dependent splitting functions. Then we use the obtained Pgq kernel to construct a low-x evolution equation for gluons that takes into account the effect of non-diagonal quark-to-gluon splittings. In order to write down a consistent equation we resum virtual corrections coming from the gluon channel and demonstrate that this implies a suitable regularization of the Pgq singularity, corresponding to a soft emitted quark. We also note that the obtained equation is in a straightforward manner generalized to a nonlinear evolution equation which takes into account effects due to the presence of high gluon densities.

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